MODELING PLASTICITY/DAMAGE USING INTERNAL STATE VARIABLES

2. MODELING PLASTICITY/DAMAGE USING INTERNAL STATE VARIABLES DOUGLAS J. BAMMANN CENTER FOR MATERIALS AND ENGINEERING SCIENCES SANDIA NATIONAL LABS/CA

3. Phenomenological Modeling Using State Variables • Kinematics • Thermodynamics • Physical basis (micromechanics) • Parameter determination • Microstructural (physical) constants • fixed state experiments • evolving state experiments • analytical solutions - simple tests - nonlinear least squares • numerical integration - optimization • Numerical implementation • Validation

4. Coleman-Gurtin applied to plasticity (Kratochvil-Dillon, Werne-Kelly, Teodosiu-Sidoroff, Perzyna,Anand ... • Assume multiplicative decomposition of deformation gradient into elastic and plastic parts (unloading elastically defines stress free or intermediate configuration - this assumption couples kinematics with constitutive model) • Assume that the free energy, defined with respect to the intermediate configuration, depends upon the elastic strain and deformation type (or defects) state variables • Determine restrictions of symmetry and invariance • Develop evolution equations for the state variables (generally motivated from the microstructural effects they represent) • Conjugate thermodynamic driving forces are defined as the derivative of the the free energy w.r.t. the state variables and are the stress like variables in the flow rule • Chosen as a result of the appearance of the product of these two terms in the dissipation inequality • Determine material parameters • Microstructural constants • Fixed state experiments • Evolving state experiments

5. The plasticity variables (those related to deviatoric plastic flow from dislocations) are motivated as follows • Evolution of variables cast in hardening minus recovery format • hardening is based upon a dislocation storage mechanism and has temperature dependence of shear modulus • dynamic recovery is motivated by cross slip and operates on same time scale as plastic flow and therefore its rate dependence is determined by the kinetics of slip • thermal recovery is motivated by diffusional climb and operates on a completely different time scale and introduces a strong rate dependence • Scalar variable represents statistically stored dislocations and gives rise to most of the hardening • Dislocations are stored inversely proportional to mean free path • Tensor variable describes geometrically necessary dislocations • Tensor variable hardens in direction of plastic flow and recovers in the direction of the current value of the tensor variable • Under constant temperature, strain rate and loading direction both variables reach steady state when hardening is balanced by recovery

6. Configurations Kinematics of continuum mechanics begins by introducing a map frommaterial space (manifold) to physical space (R3). Thus the map is from a reference configuration in material space to the current configuration in physical space. The reference configuration is not to be confused with the Lagrangian configuration which is simply a previously occupied configuration in physical space. To simplify matters we will consider the reference configuration to be the initial or Lagrangian configuration as we develop our 1D continuum x Bt X x0 BR Bt0

7. KINEMATICS Following Bilby (1956), Kroner (1960) decompose the deformation gradient into elastic and plastic parts

8. Strain can be defined with respect to any configuration based upon the Lagrange change in square length per unit square length Define a Lagrange total and plastic strain w.r.t. reference configuration Then the elastic strain in this configuration is defined as These can be mapped forward to the intermediate configuration

9. The velocity gradient is defined as usual and is naturally composed of elastic, plastic, damage and thermal parts. These can be mapped to any other configuration. Velocity gradient Notice that the plastic part of the velocity gradient is naturally defined with respect to the natural configuration as And the velocity gradient w.r.t any configuration can be split into elastic and plastic parts And then from algebra

10. The decomposition of the deformation gradient into elastic and plastic parts • Results in additional degrees of freedom • Requires specifying expressions for plastic stretching (strain rate) and plastic spin • These are easy to specify in crystal plasticity, but more difficult to motivate in phenomenological models • Most models ignore plastic spin • This can be shown to be critical in attempting to predict evolving anisotropy (from texture)

11. INTERNAL STATE VARIABLE THEORY • What are state variables? • Variable whose current value represents some observable state of the material • Can be initialized (can be measured without knowing anything about the past) • Field theory • How are kinematics related to the state variables? • Define thermodynamics with respect to specifically defined kinematic configuration • How does kinematics or the geometric structure of the state variables affect the degrees of motion of the continuum

12. The free energy of the crystal depends upon the dislocations present while the plastic deformation is governed by the transport of dislocations Y = Y0 s = s0 ep= 0 Y = Y0+DY s = s0 +Ds ep= b/2 Y = Y0 s = s0 ep= b

13. Rate of change of internal energy is the sum of the rate at which work is being done on the body, , balanced by rate at which heat is supplied to the body, h Thermodynamics - Coleman and Gurtin (1967) (Formulation will be small strain, 1D to simplify concepts) Kinematics: Thermodynamics - 1st Law 2nd Law

14. Assume that the free energy depends upon the elastic strain, the temperature, and two state variables, the elastic strain resulting from the distribution of geometrically necessary dislocations leading to kinematic hardening, gnand the elastic strain associated with the density of statistically stored dislocations giving rise to isotropic hardening, ss It follows from the 2nd law Where, and  are conjugate thermodynamic stresses

15. Then neglecting elastic heating effects and neglecting conduction for high strain rate applications, the energy balance reduces to And the dissipation inequality reduces to The kinematics of plasticity introduced degrees of freedom requiring more”constitutive” equations. Similarly, the extra degrees of freedom also requires temporal evolution equations to complete the system. This is where the physics of the smaller length scales enters!

16. Analogous to elasticity, assume a free energy of the form Then, To complete the system we need An expression for the plastic strain rate Evolution equations for the state variables   Now let’s take a very brief look at dislocation models of plasticity to determine the necessary forms

17. Statistically Stored Dislocations b Zero net burgers vector -b

18. Geometrically Necessary Dislocations Net burgers vector is 2 b b b

19. Polycrystal deforms, grains rotate, lose continuity. Geometrically necessary dislocations permit reassembly of polycrystal void overlap Geometrically or Kinematically Necessary Dislocations and/or Boundaries

20. Some Important Deformation Mechanisms - Klahn, Mukherjee, Dorn Drag Mechanisms Thermally Activated SHEAR STRESS Athermal Diffusion Controlled 200 400 600 800 TEMPERATURE K

21. Frost-Ashby deformation mechanism map

22. Example - Thermally Activated Motion (reaction-rate theory, Eyring (1936)) • Assume the number of times per second that a dislocation segment overcomes an energy barrier under the action of an applied stress can be written as a thermally activated process. •  • Q0 = height of energy barrier • bA = work done by applied stress in overcoming energy barrier or effectively lowering barrier • k = Boltzman constant •  = temperature • A = area swept in glide plane when dislocation segment moves to top of barrier • b = Burgers vector

23. Frequency of jumps backward over the barrier Net forward reaction rate If l is the length of the dislocation freed after each successful jump, and V is the volume of the crystal, the strain after each jump is And the strain rate is then Where N is the number of dislocation segments (activation sites) per unit volume

24. Force lb lb lb lb Distance along glide path A/l Force - Distance Diagram (constant rate and temperature)  lb lb- force of applied stress Lb - force of average value of spatially fluctuating long range internal stress from other dislocations lb - short range obstacle strength

25. Ono (1968) and Kocks et al. (1975) showed that a large range of obstacle shapes could be described by the following Where F is the activation energy characterizing the strength of a single obstacle. It also determines the rate sensitivity of the internal strength, .

26. Elastic collapse - stress exceeds ideal shear strength Thermally activated motion Peierls force - lattice resistance Power Law Creep Deformation Mechanisms - Frost and Ashby

27. Harper-Dorn Creep Power Law Breakdown Diffusional Flow Net plastic Flow

28. Viscous Drag Phonon and Electron

29.  Evolution of dislocation density Motivate evolution equations from Kocks-Mecking where dislocation density evolves as a dislocation storage minus recovery event. In an increment of strain dislocations are stored inversely proportional to the mean free path l, which in a Taylor lattice is inversely proportional to the square root of dislocation Density. Dislocations are annihilated or “recover” due to cross slip or climb in a manner proportional to the dislocation density A scalar measure of the stored elastic strain in such a lattice is

30. Model Development Rather than introducing several flow rules, we propose a temperature dependence for the initial value of the internal strength that emulates all of the mechanisms at a very low strain rate

31. Linear Elasticity Introduce a flow rule of the form From dislocation mechanics, (statistically stored dislocations)

32. The tensor variable is motivated by the continuum theory of dislocations (recall the nonlocal workshop where we introduced the elastic curvature as a state variable resulting in a natural gradient and natural bacck stress). Here we simplify to a local form and choose Heat conduction reduces to As dislocations are stored, the dissipation is reduced until recovery becomes dominant. Then the heat dissipated approaches the assumption that 90-95% of plastic work is dissipated as heat.

33. The tensor variable is motivated by the continuum theory of dislocations (recall the nonlocal workshop where we introduced the elastic curvature as a state variable resulting in a natural gradient and natural back stress). Here we simplify to a local form and choose For a wide range of temperatures and strain rates, plastic flow is a thermally activated strain rate. By choosing an appropriate form for the activation energy we get a flow rule of the form

34. Two parameter fit of 304L SS data at 800C for both large and small strain offset definitions of yield. For two parameters, the model reduces to rate independent bilinear hardening. Data Large offset Small offset STRESS MPa STRAIN

35. Recovery included for the same compression curve. In this case the model accurately captures both the hardening and recovery through the isotropic hardening variable . STRESS MPa data s  STRAIN

36. The small strain fit can be improved by including the short transient a which saturates at small strains as a function of its hardening and recovery parameters

37. Six parameter fit of 304L SS compression data with only the long transient k but including the effects of rate dependence of yield through the parameters V and f . The strain dependent rate effect is captured by the static recovery parameter Rsk . model10[1/s] model10-1[1/s] k 10[1/s] k 10-1[1/s]

38. Five parameter fit of 304l SS compression curve including the short transient a . This fit will more accurately capture material response during changes in load path direction

39. Model prediction for 304L stainless steel tension tests or 304L stainless steel is depicted in Figure 1.

40. Model prediction of compression tests and compression reload demonstrating temperature and history effects. The 800C test was quenched after being strained to 23% and reloaded at room temperature. The temperature history effect is demonstrated by the reload curve being much softer than the 20C curve.

41. 1 2 a 3 1 2 3 Kinematic vs. Isotropic Hardening • If all hardening occurs uniformly by statistically stored dislocations, (and the texture is random), the yield surface would grow isotropically “the same in every direction, independent of the direction of loading”. The radius of the yield surface, is given by k, the internal strength of the material. • This type of loading is illustrated in the figures. The material deforms elastically and the stress increases linearly until the initial yield surface is reached and the material hardens and the yield surface grows until unloading begins at point 1. Upon reversal of load the material deforms elastically until point 3 is reached. • If geometrically necessary dislocations form pileups at grain boundaries (small effect) or at particles (larger effect), the material exhibits an apparent softening upon load reversal • To model this, the yield surface is allowed to translate to the same stress point 1 (red surface). Now upon load reversal, plastic flow begins at point 2. Real material would begin a combination of these two exaggerated figures. This is a short transient and a represents the center of the yield surface. • In some cases, we used to use a as long transient to model texture effects. But now we introduce a structure tensor for this effect. k

42. 1 1 2 2 The proportion of kinematic and isotropic hardening can be determined from reverse loading tests • To satisfy yield at both unloading and reverse yielding (assuming state variables don’t change during elastic unloading) • Then the kinematic and isotropic hardening proportions can be determined as

43. The offset used to to determine yield has a large effect upon the proportion of kinematic to isotropic hardening • A small offset definition of yield stress (such as a strain of 0.005 %) results in a greater proportion of kinematic hardening • A larger offset (the standard 0.2 % strain) will generally result in a prediction of a domination isotropic hardening • The truth (experimental data) is most closely approximated by the smallest offset and is most important in problems involving unloading at small strains

44. J. Hodowany’s Ph.D. dissertation (Caltech, 1997) showed quantitative measurements of the conversion of plastic work into heat as a function of plastic strain Hodowany’s measurements indicate that in the early stages of deformation, the amount of plastic work converted to heat may be quite small As deformation progresses, the heat conversion more nearly equals the plastic work rate

45. Results of J. Hodowany • (Ph.D. dissertation, Caltech, 1997) • Kolsky (split Hopkinson) bar arrangement • High speed measurements of • Temperature (IR detector) • Plastic work (strain gages on input and output bars)

46. Primary results involve a parameter b which is the fraction of plastic work converted to heat

47. Fraction of plastic work converted to heat

48. Fraction of plastic work converted to heat for 2024 Al

49. Simulation for a “fully annealed” material

50. Simulation for an initially hardened material